Improving coverage of rectangular confidence interval

by Gogtas, Hakan

Abstract (Summary)

To find a better confidence region is always of interest in statistics. One way to find better confidence regions is to uniformly improve coverage probability over the usual confidence region while maintaining the same volume. Thus, the classical spherical confidence regions for the mean vector of a multivariate normal distribution have been improved by changing the point estimator for the parameter
In 1961, James and Stein found a shrinkage estimator having total mean square error, TMSE, smaller than that of the usual estimator. In 1982, Casella and Hwang gave an analytical proof of the dominance of the confidence sphere which uses the James Stein estimator as its center over the usual confidence sphere centered at the sample mean vector. This opened up new possibilities in multiple comparisons
This dissertation will focus on simultaneous confidence intervals for treatment means and for the differences between treatment means and the mean of a control in one-way and two-way Analysis of Variance, ANOVA, studies. We make use of Stein-type shrinkage estimators as centers to improve the simultaneous coverage of those confidence intervals. The main obstacle to an analytic study is that the rectangular confidence regions are not rotation invariant like the spherical confidence regions
Therefore, we primarily use simulation to show dominance of the rectangular confidence intervals centered around a shrinkage estimator over the usual rectangular confidence regions centered about the sample means. For the one-way ANOVA model, our simulation results indicate that our confidence procedure has higher coverage probability than the usual confidence procedure if the number of means is sufficiently large. We develop a lower bound for the coverage probability of our rectangular confidence region which is a decreasing function of the shrinkage constant for the estimator used as center and use this bound to prove that the rectangular confidence intervals centered around a shrinkage estimator have coverage probability uniformly exceeding that of the usual rectangular confidence regions up to an arbitrarily small epsilon when the number of means is sufficiently large. We show that these intervals have strictly greater coverage probability when all the parameters are zero, and that the coverage probability of the two procedures converge to one another when at least one of the parameters becomes arbitrarily large
To check the reliability of our simulations for the one-way ANOVA model, we use numerical integration to calculate the coverage probability for the rectangular confidence regions. Gaussian quadrature making use of Hermite polynomials is used to approximate the coverage probability of our rectangular confidence regions for n=2, 3, 4. The difference in results between numerical integration and simulations is negligible. However, numerical integration yields values slightly higher than the simulations
A similar approach is applied to develop improved simultaneous confidence intervals for the comparison of treatment means with the mean of a control. We again develop a lower bound for the coverage probability of our confidence procedure and prove results similar to those that we proved for the one-way ANOVA model.
We also apply our approach to develop improved simultaneous confidence intervals for the cell means for a two-way ANOVA model. We again primarily use simulation to show dominance of the rectangular confidence intervals centered around an appropriate shrinkage estimator over the usual rectangular confidence regions. We again develop a lower bound for the coverage probabilities of our confidence procedure and prove the same results that we proved for the one-way model